Finding Near Rank Deficiency in Matrix Products

Abstract

This paper gives a theorem characterizing approximately minimal norm rank one perturbations E and F that make the product (A + E)(B + F)T rank deficient. The theorem is stated in terms of the smallest singular value of a particular matrix chosen from a parameterized family of matrices by solving a nonlinear equation. Consequently, it is analogous to the special case of the Eckhart-Young theorem describing the minimal perturbation that induces an order one rank deficiency. While the theorem does not naturally extend to higher order rank deficiencies, it can be used to compute a complete orthogonal product decomposition to give improved practical reliability in revealing the numerical rank of ABT.